Proof.
We shall show that the change of variable
,
provides a conformal map from the z-plane to the s-plane that takes the
region
to the region
re. The general formula for a
bilinear conformal mapping of functions of a complex variable is given by

(C.85)

In general, a bilinear transformation maps circles and lines into circles
and lines [83]. We see that the choice of three specific points
and their images determines the mapping for all and .
We must have that the imaginary axis in the s-plane maps to the unit circle
in the z-plane.
That is, we may determine the mapping by three points of the form
and
.
If we predispose one such mapping by choosing the pairs
and
, then we are left with
transformations of the form

or

(C.86)

Letting be some point on the imaginary axis,
and be some point
on the unit circle, we find that

which gives us that is real. To avoid degeneracy, we require
, and this translates to finite and
nonzero. Finally, to make the unit disk map to the left-half s-plane,
and must have the same sign in which case .

There is a bonus associated with the restriction that be real which
is that

(C.87)

We have therefore proven

Property.

PR PR

where is any positive real number.

The class of mappings of the form Eq.(C.85) which take the exterior of
the unit circle to the right-half plane is larger than the class
Eq.(C.86). For example, we may precede the transformation
Eq.(C.86) by any
conformal map which takes the unit disk to the unit disk, and these
mappings have the algebraic form of a first order complex allpass
whose zero lies inside the unit circle.

(C.88)

where is the zero of the allpass and the image (also pre-image)
of the origin, and is an angle of pure rotation. Note that
Eq.(C.88) is equivalent to a pure rotation, followed by a real
allpass substitution ( real), followed by a pure rotation. The
general preservation of condition (2) in Def. 2forces the real axis
to map to the real axis. Thus rotations by other than are
useless, except perhaps in some special cases. However, we may
precede Eq.(C.86) by the first order real allpass
substitution

real

which maps the real axis to the real axis. This leads only to the composite
transformation,

which is of the form Eq.(C.86) up to a minus sign (rotation by
). By inspection of Eq.(C.85), it is clear that sign negation
corresponds to the swapping of points and , or and .
Thus the only extension we have found by means of the general disk to
disk pre-transform, is the ability to interchange two of the three
points already tried. Consequently, we conclude that the largest
class of bilinear transforms which convert functions positive real in
the outer disk to functions positive real in the right-half plane is
characterized by

(C.89)

Riemann's theorem may be used to show that Eq.(C.89) is also the
largest such class of conformal mappings. It is not essential,
however, to restrict attention solely to conformal maps. The
pre-transform
, for example, is not conformal and yet PR
is preserved.

The bilinear transform is one which is used to map analog filters into
digital filters. Another such mapping is called the matched
transform [362]. It also preserves the positive real
property.

Property. is PR if is positive real in the analog
sense, where is interpreted as the sampling period.

Proof. The mapping
takes the right-half -plane to
the outer disk in the -plane. Also is real if is
real. Hence PR implies PR. (Note, however, that
rational functions do not in general map to rational
functions.)

These transformations allow application of the large
battery of tests which exist for functions positive real in the right-half
plane [524].